It also doesn't tell you the elasticity. How much the demand would change if the price were increased by a unit increment. The only way to truly know that is... to change the price. The CPI can only measure how the price changed, which is a dependent variable, not an independent variable. You may be able to infer the elasticity, but it is at best a weak inference.
CPI tells you how much prices have changed, it doesn't explain the source of the changes (and, particularly relevantly, it doesn't measure isolate effects of changes in the supply of dollars chasing the good -- that would be attributed to the degree of elasticity -- vs. other influences on the price of an item.)